

In [1]:

    
%matplotlib inline
%load_ext autoreload
%autoreload 2
from CO2simulation import CO2simulation
import runCO2simulation as simCO2
import matplotlib.pyplot as plt
import numpy as np
import visualizeCO2 as vco2
import matplotlib as mpl
from display_notebook import css_styling
from IPython.display import Image
mpl.rcdefaults()
mpl.matplotlib_fname()
css_styling() # suppress matplotlibrc









    Out[1]:

Scalable Kalman filtering for Temporal-Spatial Analysis

### Seismic monitoring of CO$_2$
- State estimation
- Time-series data
### Methodology
- Kalman filter
- Scalability
### Result Visit my GitHub page!

Track a CO$_2$ Plume from Sensor Measurements

Objective: monitor a CO$_2$ plume for $5$ days resulted from injecting $300$ tons of CO$_2$ at a depth of $1657m$.

The sensor measures the travel time of a seismic signal from a source to a receiver.
The presence of CO$_2$ slows down the seismic signal and causes travel time delay along a ray path.
Goal: interpret the changes in seismic signals into maps of moving CO$_2$ plume.



In [2]:

    
from IPython.html.widgets import interact, interactive, fixed
from IPython.html.widgets import FloatSlider
from CO2simulation import CO2simulation
def plot_CO2plume(time):
    import param as param
    CO2 = CO2simulation(param)
    x = CO2.extract_state(int(time/3))
    data = CO2.extract_data(int(time/3))
    fig_setting = vco2.getImgParam(param)
    vco2.plotCO2_data_map(x, data, 0, 20, fig_setting)
    plt.show()
    
interact(plot_CO2plume,
         time = FloatSlider(value=0, min=0, max=120));









    



/Users/yueli/anaconda/lib/python2.7/site-packages/matplotlib/collections.py:590: FutureWarning: elementwise comparison failed; returning scalar instead, but in the future will perform elementwise comparison
  if self._edgecolors == str('face'):

Kalman filtering

Linear dynamic system, estimation, inference
Two phases:
- Prediction
- Update

Scalability Issues

Computationally prohibitive for large-scale problem
Store and update covariance matrix ($N^2$) at each Kalman step
Computational cost grows quadratically with $N$
$80$ days for a typical problem size ~1 million

Resolution	Low	Medium	High
State dimension	N = 3245	N = 3245 x 4	N = 3245 x 16
Run time	1.2 min	19 minutes	4.4 hours
Storage cost	100 MB	1331 MB	20 G

Scalable Kalman filtering

$\mathcal{O}(N)$: a linear computational complexity algorithm
Conventional dimension reduction algorithms (e.g.,PCA) truncates information
Lossless compression for a kernel covariance matrix
Hierarchical matrices approach efficiently explore the structure of a kernel matrix



In [4]:

    
import visualizeCO2 as vCO2
vCO2.scale_barplot()

Results

                                    TRUE                             HIKF

Filter Design

By choosing an appropriate $Q/R$ ratio to optimize the filter preformance



In [3]:

    
# theta controls the trade-off between data misfit and variance in final estimates
def plot_CO2maps(theta):
    import param
    CO2 = CO2simulation(param)
    param.theta = (theta,1e-5)
    hikf, x_kf, cov_kf = simCO2.CO2_filter(CO2, param)
    fig_setting = vco2.getImgParam(param)
    vco2.plotCO2map(x_kf,cov_kf,fig_setting)
    plt.show()
    print "Theta                  Variance"
    print " %f              %f" % (theta,np.sum(cov_kf[-1]))
    
interact(plot_CO2maps,
         theta = FloatSlider(value=1.14, min=1.14e-3, max=1.14e1, step=1));









    












    



Theta                  Variance
 4.001140              0.292195

Summary

Efficient tools to interprete the time-series data recorded in the seismic sensors into spatial maps of a moving CO$_2$ plume, a problem very similar to CT scanning widely used in medical imaging.